ON THE COMPUTATIONAL CHAOS IN THE FINITE DIFFERENCE SCHEMES WITH NON-UNIFORM GRIDS
Zeng Qing-cun Li Rong-feng(Institute of Atmospheric Physics, Academia Sinica)
The problems of computational chaos in the finite difference schemes with non-uniform grids or with two sets of grids matching each other are investigated by using the physical analytic method. It is concluded that false reflective waves with wavelength of 2 grid-sizes are generated in the area with gross grids as the disturbances propagate from the gross grids to the fine grids; on the other hand, if the incident wave comes from the area with fine grids false reflective waves are generated in the area with fine grids and if its frequency exceeds the critical value (so-called "the ultra high frequency condition"), and almost all the wave energy is reflected as the false reflective wave, thus results in serious computational chaos. Therefore, caution must be taken to avoid "the ultra high frequency condition" in the design of computational scheme.In this paper, a comparison has been made among the three methods for design of non-uniform grid schemes, i.e. (a) the method of coordinate transformation, (b) the method of weighted averaging, and (c) Isaacson method. It is indicated that under the condition of non-existence of "the ultra high frequency condition" all three methods give only small false-reflectivity with the best result obtained by using (b) if the wavelength is longer than 6 ds, on the other hand a small false-reflectivity can be obtained only by using (b) if the wavelength is very short.